O centro das álgebras envolventes universais de álgebras de Lie nilpotentes em caraterística prima

Abstract

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $\mathbb{F}$ of prime characteristic $p$ and let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$. We know from the literature that the center $Z(\mathfrak{g})$ of the enveloping algebra $U(\mathfrak{g})$ is an integrally closed domain. In this thesis, we describe $Z(\mathfrak{g})$ for all indecomposable nilpotent Lie algebras of dimension less than or equal to $6$ over a field $\mathbb{F}$ of prime characteristic $p$ assuming that $\mbox{cl}(\mathfrak{g})\leq p$. We present examples where $Z(\mathfrak{g})=Z_{p}(\mathfrak{g})$, where $Z_{p}(\mathfrak{g})$ is the $p$-center of $U(\mathfrak{g})$. However, we found cases where $Z(\mathfrak{g})\neq Z_{p}(\mathfrak{g})$. In these cases, we will deal with integral extensions $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]\subseteq Z(\mathfrak{g})$ where $z_{1},\ldots, z_{s}\in Z(\mathfrak{g})\setminus Z_{p}(\mathfrak{g})$. When $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]\subseteq Z(\mathfrak{g})$ is an integral extension whose fraction fields coincide, for equality $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]=Z(\mathfrak{g})$ to occur, it is sufficient that $Z_{p}(\mathfrak{g})[z_{1},\ldots z_{s}]$ is an integrally closed domain. A good part of our job is to show this property. At this point, we need the concepts of regular rings and Cohen-Macaulay rings. We denote by $S(\mathfrak{g})$ the symmetric algebra of $\mathfrak{g}$. In characteristic zero, the spaces $U(\mathfrak{g})$ and $S(\mathfrak{g})$ are isomorphic $\mathfrak{g}$-modules with respect to the adjoint representation. The set of invariants of the $\mathfrak{g}$-module $U(\mathfrak{g})$ is its center $Z(\mathfrak{g})$ and the set of invariants of the $\mathfrak{g}$-module $S(\mathfrak{g})$ is denoted by $S(\mathfrak{g})^{\mathfrak{g}}$. In this thesis, we describe the algebra of invariants $S(\mathfrak{g})^{\mathfrak{g}}$ for all indecomposable nilpotent Lie algebras of a dimension less than or equal to $6$ over a field $\mathbb{F}$ of prime caracteristic $p$ with $\mbox{cl}(\mathfrak{g})\leq p$. Furthermore, we show the existence of an algebra isomorphism between $Z(\mathfrak{g})$ and $S(\mathfrak{g})^{\mathfrak{g}}$. We also describe $Z(\mathfrak{g})$ and $S(\mathfrak{g})^{\mathfrak{g}}$ for the standard filiform Lie algebras of dimension up to $6$ in prime characteristic $p$. For these algebras, in general, it is not an easy task to determine explicit generators for $Z(\mathfrak{g})$ and $S(\mathfrak{g})^{\mathfrak{g}}$ because of the complexity of their generators and their relations. Keywords: nilpotent Lie algebras, universal enveloping algebra, center, Poisson center, Cohen-Macaulay rings.